It's not something that
could be screen-grabbed, even if I was concerned enough to bother.
I'll either plot a course, I'll see the messages telling me my FC is going to jump in 10 minutes, or whatever, and I'll even hear the "Departure Imminent" messages if I'm aboard, then I'll go and do something else, travel to the system which is the destination of my FC and find it's not there.
I'll check the FC UI and find that it's still sitting in it's original system, and will (probably) cooperate if I order it to jump again.
The only way I
could screen-cap this issue would be to take screenies
every time I issue a jump order and then take another one after the FC
doesn't actually make the planned jump.
Equally, I can't screen-cap the cooldown not commencing because it only ever seems to happen when I'm
not paying close attention to what my FC is doing.
For example, I'll have a ship on my FC that needs it's FSD engineering.
I'll be off collecting mat's and I'll jump my FC to Deciat.
I'll spend a couple of hours collecting mat's, dock at my FC (in Deciat), swap to the required ship so I can get the FSD engineered, attempt to plot another FC jump and find I can't because it's still in cooldown - even though it has, apparently, been sitting around in Deciat for the best part of a couple of hours.
As for the thing about player-ships being left behind when an FC jumps, I think that's a fairly well-accepted phenomenon now.
It tends to happen when you're
not in-game so, again, it's impossible to screencap in any meaningful way.
It's always (for me) at the end of a session, where I'll dock at my FC, order it to jump somewhere, log off, log back in the next day and find my ship is floating in space, in the origin system, and I need to travel to meet my FC in the destination system.